Almost Periodicity in Exponential Dichotomous Linear Dynamics with Piecewise Constant Argument

Abstract

The paper examines an inhomogeneous system of differential equations with generalized piecewise constant arguments. The focus of the research is on the almost periodicity of both the perturbation and the solution. The system demonstrates exponential dichotomy, indicating complex asymptotic behavior among neighboring motions. A notable methodological aspect of the analysis is the use of Green's function to construct a unique almost periodic solution. Additionally, a comparison with previous results has been made. An illustrative example with numerical visualizations is provided to clarify this concept.